In Michigan, now don't get confused with me telling you how it is nation wide here in the states because it isn't. So check with your Canadian authorities. But in Michigan, I can call my business anything I want, go down to the country clerks office and pay $35 to get my business licsence and a piece of paper that says I'm doing business as "Your Fav ProdCo". I take that paper to the bank and open up a checking account and apply for a credit/debit card.
Now if someone in LA sees my web site and thinks, "WOW cool name wish it was mine." Does some checking and wants to be a real a$$hole about it, they can go get a lawyer, apply for a national or even world wide doing business as "Your Fav ProdCo" and suddenly I'm fooked. I have to change my name.
Bottom line is, people can be a$$holes and that's why you need to talk with an intellectual property rights lawyer where you live.